# Flu Math

Image: GARY ZAMCHICK

A recent short article in Scientific American Mind looked at the following question: suppose you have a 5 percent chance of dying from a flu vaccine but a 10 percent chance of contracting and dying of the flu when an epidemic strikes. Do you take the flu shot? Surprisingly often, people do not. (See "Which Flu Risk Would You Take?," by Nicole Garbarini, in Head Lines, Scientific American Mind, Aug./Sept. 2006).

This apparent irrationality is commonly attributed to "omission bias"-people often prefer inaction to action, even if inaction carries some greater risk. But I think there are other reasons, too. For one thing, authorities commonly have a bias toward action, if only to justify their own existence. If so, the consequences of the flu may be overstated or the risks of the vaccine are understated.

But even if the listed probabilities are to be believed and the bias toward inaction is corrected, there is the observation that the risk of being unvaccinated decreases if most other people take the flu vaccine.

Suppose in fact that your likelihood of dying from the flu when unvaccinated goes down according to the following formula: if a fraction f (excluding you) of the population takes the flu shot, then your probability of dying from the flu is just (1-f)*10%. For example, if 65 percent of the people take it and you are among the 35 percent who do not, then the probability of your dying from the flu is only 0.35 * 10% = 3.5%.

Warm-Up:

Suppose a government official could require 60 percent of the people to take the vaccine. Then what would be the average risk of death for the entire population due to the flu?

Solution to Warm-Up:

Recall that if the official doesn't allow anyone to take the vaccine, then the death toll is 10 percent. If the official forces everyone to take it, the death toll is 5 percent. If 60 percent take the shot, then those people have a 5 percent chance of dying, but the others have only a 40 percent chance of dying from the flu, so the death toll among them is 4 percent. The overall death toll therefore is (0.6 * 5%) + (0.4 * 4%) = 4.6%.

1. As long as the government is in a compelling mode, what fraction should be required to take the vaccine to minimize the average risk overall?

On the other hand, suppose that the government feels it cannot compel people to take the shot. Instead, the government offers each person the flu shot in turn. If a person refuses, then there is no second chance. Each person knows how many people took the flu shot among those already offered. Each person believes and knows that everyone else believes the government's risk figures (5 percent if a person takes it; 10 percent modified by the f formula above if a person doesn't). Each person will take the flu shot if and only if it helps him or her. There is no regard for the greater good.

2. Under those conditions, what percentage of people will take the flu shot?

The government looks at the results and decides that a little benevolent disinformation is in order. That is, the government will inflate the publicized risk of death from the flu to a number R that is greater than 10 percent, but the decrease in risk will also be adjusted to use R instead of 10 percent-that is, to (1-f)*R. The exaggerated risk strategy is a carefully guarded secret, so everyone believes the government and knows that everyone else does too.

3. Again, the government will compel nobody. To which percentage should the risk of disease be inflated to achieve the optimum vaccination level you determined in your answer to question 1?

Disclaimer: The actual death rates associated with the flu and flu vaccines are typically far smaller than the numbers used in these examples. The scenario of benevolent disinformation is purely invented. Purely.

By Dennis Shasha, professor of computer science at the Courant Institute of New York University. His latest puzzle book, The Puzzler's Elusion, appeared this past spring.

View
1. 1. -larry 07:31 PM 5/11/09

What a crock of confusing mathematics! The percentage of death from having the flu does not change... e.g., .6 x 5% plus .4 x 10% yields the overall death rate (in the case that 60% of the population is vaccinated). That answer is "7%" by the way, not 4.6%--how do you achieve a number less than the 'best' case scenario when everyone is vaccinated? That should be the first 'sanity check' on the math. This is straight forward...it's no wonder that people are confused, especially when a respected scientific magazine presents this.
There are additional considerations. What percentage of the population will get the flu if an epidemic does occur. In your example, if only 50% of the population gets the flu then the real risk of death from the flu is the same as getting the vaccine50% chance of getting the flu times 10% chance of death if you do get it or 5% of the unvaccinated population.

2. 2. Tigerlsu 02:31 AM 5/20/09

I wonder how does the author get the death from 10% to 4%?

3. 3. MarkusD 03:41 PM 5/21/09

Hey guys,

The author said that the odds of contracting AND dying from the disease are 10%. This number is only valid when zero percent of the population is vaccinated. As the percentage of vaccinations approaches 100% the probability of non-vaccinated people contracting and dying from the disease approaches zero. I'm not sure it would be linear in real live but it makes intuitive sense at the end points. For example, if 99 out of 100 people were vaccinated and you were not, your odds of getting and dying from the disease would be really low (1% in this case) simply because there would be no one to pass it on to you.

That's how the author gets the overall death rate to less than 5%. Once the vaccination rate passes 50% of the population the odds of a non-vaccinated person dying becomes less then 5%.

Anyway, the optimal vaccination rate is 75%. At this point, the overall odds of dying from the disease is 4.375%. The odds of vaccinated people dying are still 5% but the odds of the non-vaccinated people dying are only 2.5%.

4. 4. MarkusD 03:44 PM 5/21/09

Hey guys,

The author said that the odds of contracting AND dying from the disease are 10%. This number is only valid when zero percent of the population is vaccinated. As the percentage of vaccinations approaches 100% the probability of non-vaccinated people contracting and dying from the disease approaches zero. I'm not sure it would be linear in real live but it makes intuitive sense at the end points. For example, if 99 out of 100 people were vaccinated and you were not, your odds of getting and dying from the disease would be really low (1% in this case) simply because there would be no one to pass it on to you.

That's how the author gets the overall death rate to less than 5%. Once the vaccination rate passes 50% of the population the odds of a non-vaccinated person dying becomes less than 5%.

Anyway, the optimal vaccination rate is 75%. At this point, the overall odds of dying from the disease are 4.375%. The odds of vaccinated people dying are still 5% but the odds of the non-vaccinated people dying are only 2.5%.

You must sign in or register as a ScientificAmerican.com member to submit a comment.
Click one of the buttons below to register using an existing Social Account.

## More from Scientific American

• News | 6 hours ago | 2

### Jupiter's Moon Europa Spotted Spouting Water

• Reuters | 6 hours ago | 3

### U.S. 2012 Model-Year Vehicles Hit Record Fuel Efficiency

• Guest Blog | 6 hours ago

### How Much Nature Do We Have to Use?

• Cross-Check | 7 hours ago

### Celebrities Should Inform Women about Risks as Well as Benefits of Mammograms

• Observations | 7 hours ago

## Latest from SA Blog Network

• ### 5 Signs of Life on Mars - The Countdown, Episode 37

The Countdown | 4 hours ago
• ### Dog Farts Part 2: How to Make Dog Farts Less Stinky

MIND
Dog Spies | 5 hours ago
• ### How Much Nature Do We Have to Use?

Guest Blog | 6 hours ago
• ### Celebrities Should Inform Women about Risks as Well as Benefits of Mammograms

Cross-Check | 7 hours ago
• ### Duck-billed Dinosaur Had Bizarre Rooster's Comb, Mummy Find Reveals

Observations | 7 hours ago

## Science Jobs of the Week

Flu Math

X

Give a 1 year subscription as low as \$14.99

X

X

###### Welcome, . Do you have an existing ScientificAmerican.com account?

No, I would like to create a new account with my profile information.

X

Are you sure?

X